Q. 28 - Calculus

Question

In Exercises 27–32, functions x = x(u, v) and y = y(u, v) are given that determine transformations from an XY-coordinate system to a UV-coordinate system in R2. Use these functions to determine a region in the XY-plane that has the image specified for the given values of u and v, and find the Jacobian of the transformation. 

$x = 3u + 4v and y = 4u - 3v for 1 \le u \le 3 and 1 \le v \le 5$

Step-by-Step Solution

Answer

The Jacobian is equal to $-25$.

Step 1: Given information

The functions are,

$x = 3u + 4v and y = 4u - 3v for 1 \le u \le 3 and 1 \le v \le 5$

Step 2: Find the Jacobian

The Jacobian of transformation is computed as,

$$\begin{gathered} \frac{\partial (x, y)}{\partial (u,v)}=det\left[\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v}& \frac{\partial y}{\partial v}\end{array}\right] \\ \frac{\partial (x, y)}{\partial (u,v)}=det\left[\begin{array}{cc}3& 4\\ 4& -3\end{array}\right] \\ \frac{\partial (x, y)}{\partial (u,v)}=-9-16 \\ \frac{\partial (x, y)}{\partial (u,v)}=-25 \end{gathered}$$